kinetic isotope effect

The kinetic isotope effect is a difference in the rate of a chemical reaction when an atom in one of the reactants is replaced by one of its isotopes. An isotopic substitution will greatly modify the measured reaction rate when the isotopic replacement is in a chemical bond that is broken or formed in the rate-determining step. In such a case, the rate change is termed a primary isotope effect. When the substitution is not involved in the rate-determining step, one may still observe a smaller rate change, termed a secondary isotope effect. Thus, the magnitude of the kinetic isotope effect can be used to elucidate the reaction mechanism. Isotopic rate changes are most pronounced when the relative mass change is greatest. For instance, changing a Hydrogen atom to Deuterium represents a 100% increase in mass, whereas in replacing Carbon-12 with Carbon-13, the mass increases by only 8%. The rate of a reaction involving a C-H bond is typically 6 to 10 times faster than the corresponding C-D bond, whereas a ¹²C reaction is only ~1.04 times faster than the corresponding ¹³C reaction (even though, in both cases, the isotope is one atomic mass unit heavier). Isotopic substitution can modify the rate of reaction in a variety of ways. In many cases, the rate difference can be rationalized by noting that the mass of an atom affects the vibration frequency of the chemical bond(s) that it forms, even if the electron configuration is nearly identical. Heavier atoms will (classically) lead to lower vibration frequencies, or, viewed quantum mechanically, will have lower zero-point energy. With a lower zero-point energy, more energy must be supplied to break the bond, resulting in a higher activation energy for bond cleavage, which in turn lowers the measured rate (see, for example, the Arrhenius equation). In some cases, an additional rate enhancement is seen for the lighter isotope, due to quantum mechanical tunnelling. This is typically only observed for H atoms, which are light enough to exhibit significant tunnelling. In still other cases, the rate change may be due to subtle differences in the electronegativity of the two isotopes.

Mathematical details

To see why it is the relative mass difference (and not absolute mass difference) that matters for the kinetic isotope effect, note that the fundamental vibrational frequency of a chemical bond between atom A and B (denoted by $\nu$ ) is, when approximated by a harmonic oscillator:

\nu = \frac{1}{2 \pi} \sqrt{\frac{k}{\mu}}

where $k$ is the spring constant for the bond, and $\mu$ is the reduced mass of the A-B system:

\mu = \frac{m_A m_B}{m_A + m_B}

(

m_i

is the mass of atom

i

). Quantum mechanically, the energy of the

n

-th level of a harmonic oscillator is given by:

E_n = h \nu \left ( n + \frac{1}{2} \right )

Thus, the zero-point energy (

n

= 0) will decrease as the reduced mass increases. With a lower zero-point energy, more energy is needed to overcome the activation energy for bond cleavage. Since it is the reduced mass (not absolute mass) that enters into the equations, it is clear that larger relative mass changes will lead to more noticeable isotope effects.

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